- Research Article
- Open Access
- Published:
An impulsive nonlinear singular version of the Gronwall-Bihari inequality
Journal of Inequalities and Applications volume 2006, Article number: 84561 (2006)
Abstract
We find bounds for a Gronwall-Bihari type inequality for piecewise continuous functions. Unlike works in the prior literature, here we consider inequalities involving singular kernels in addition to functions with delays.
References
- 1.
Bainov DD, Hristova SG: Impulsive integral inequalities with a deviation of the argument. Mathematische Nachrichten 1995, 171: 19–27.
- 2.
Bainov DD, Simeonov PS: Systems with Impulse Effect: Theory and Applications. Ellis Horwood, Chichister; 1989.
- 3.
Bainov DD, Simeonov PS: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic, Dordrecht; 1992:xii+245.
- 4.
Butler G, Rogers T: A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations. Journal of Mathematical Analysis and Applications 1971,33(1):77–81. 10.1016/0022-247X(71)90183-1
- 5.
Hristova SG: Nonlinear delay integral inequalities for piecewise continuous functions and applications. Journal of Inequalities in Pure and Applied Mathematics 2004,5(4):1–14. article 88 article 88
- 6.
Kirane M, Tatar N-E: Global existence and stability of some semilinear problems. Archivum Mathematicum (Brno) 2000,36(1):33–44.
- 7.
Kirane M, Tatar N-E: Convergence rates for a reaction-diffusion system. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2001,20(2):347–357.
- 8.
Krylov NN, Bogolyubov NN: Introduction to Nonlinear Mechanics. Izd. Acad. Sci. Ukr. SSR, Kiev; 1937.
- 9.
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, New Jersey; 1989:xii+273.
- 10.
Mazouzi S, Tatar N-E: Global existence for some integro-differential equations with delay subject to non-local conditions. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2002,21(1):249–256.
- 11.
Medved M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. Journal of Mathematical Analysis and Applications 1997,214(2):349–366. 10.1006/jmaa.1997.5532
- 12.
Medved M: Singular integral inequalities and stability of semilinear parabolic equations. Archivum Mathematicum (Brno) 1998,34(1):183–190.
- 13.
Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, California; 1998:x+611.
- 14.
Samoilenko AM, Perestyuk NA: Stability of solutions of differential equations with impulse effect. Differential Equations 1977,13(11):1981–1992.
- 15.
Samoilenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, New Jersey; 1995:x+462.
- 16.
Tatar N-E: Exponential decay for a semilinear problem with memory. Arab Journal of Mathematical Sciences 2001,7(1):29–45.
- 17.
Zhang W, Agarwal RP, Akin-Bohner E: On well-posedness of impulsive problems for nonlinear parabolic equations. Nonlinear Studies 2002,9(2):145–153.
Author information
Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tatar, NE. An impulsive nonlinear singular version of the Gronwall-Bihari inequality. J Inequal Appl 2006, 84561 (2006). https://doi.org/10.1155/JIA/2006/84561
Received:
Revised:
Accepted:
Published:
Keywords
- Continuous Function
- Type Inequality
- Prior Literature
- Piecewise Continuous Function
- Singular Kernel